Change of the Adiabatic Invariant due to Separatrix Crossing

When a parameter in the Hamiltonian of a one-degree-of-freedom oscillator is slowly varied at rate $\epsilon$, an adiabatic invariant exists which is conserved to all orders in $\epsilon$, except on phase-space orbits which cross a separatrix. In the present work, the change in the adiabatic invariant due to a separatrix crossing is given to order $\epsilon$ for a wide class of Hamiltonian systems. This result is applied to the special case of a charged particle moving under the influence of an electrostatic wave with slowly varying amplitude and frequency.